Estimation of Dynamic Causal Effects
(SW Chapter 13)
A dynamic causal effect is the effect on Y of a change in
X over time.
For example:
 The effect of an increase in cigarette taxes on cigarette
consumption this year, next year, in 5 years;
 The effect of a change in the Fed Funds rate on
inflation, this month, in 6 months, and 1 year;
 The effect of a freeze in Florida on the price of orange
juice concentrate in 1 month, 2 months, 3 months…
23-1
The Orange Juice Data
(SW Section 13.1)
Data
 Monthly, Jan. 1950 – Dec. 2000 (T = 612)
 Price = price of frozen OJ (a sub-component of the
producer price index; US Bureau of Labor Statistics)
 %ChgP = percentage change in price at an annual rate,
so %ChgPt = 1200ln(Pricet)
 FDD = number of freezing degree-days during the
month, recorded in Orlando FL
o Example: If November has 2 days with low temp <
32o, one at 30o and at 25o, then FDDNov = 2 + 7 = 9
23-2
23-3
Initial OJ regression
%ChgP t = -.40 + .47FDDt
(.22) (.13)
 Statistically significant positive relation
 More/deeper freezes, price goes up
 Standard errors: not the usual – heteroskedasticity and
autocorrelation-consistent (HAC) SE’s – more on this
later
 But what is the effect of FDD over time?
23-4
Dynamic Causal Effects
(SW Section 13.2)
Example: What is the effect of fertilizer on tomato yield?
An ideal randomized controlled experiment
 Fertilize some plots, not others (random assignment)
 Measure yield over time – over repeated harvests – to
estimate causal effect of fertilizer on:
o Yield in year 1 of expt
o Yield in year 2, etc.
 The result (in a large expt) is the causal effect of
fertilizer on yield k years later.
23-5
In time series applications, we can’t conduct this ideal
randomized controlled experiment:
 We only have one US OJ market ….
 We can’t randomly assign FDD to different replicates
of the US OJ market (?)
 We can’t measure the average (across “subjects”)
outcome at different times – only one “subject”
 So we can’t estimate the causal effect at different
times using the differences estimator
23-6
An alternative thought experiment:
 Randomly give the same subject different treatments
(FDDt) at different times
 Measure the outcome variable (%ChgPt)
 The “population” of subjects consists of the same
subject (OJ market) but at different dates
 If the “different subjects” are drawn from the same
distribution – that is, if Yt,Xt are stationary – then the
dynamic causal effect can be deduced by OLS
regression of Yt on lagged values of Xt.
 This estimator (regression of Yt on Xt and lags of Xt) s
called the distributed lag estimator.
23-7
Dynamic causal effects and the distributed lag model
The distributed lag model is:
Yt = 0 + 1Xt + … + pXt–r + ut
 1 = impact effect of change in X = effect of change in
Xt on Yt, holding past Xt constant
 2 = 1-period dynamic multiplier = effect of change in
Xt–1 on Yt, holding constant Xt, Xt–2, Xt–3,…
 3 = 2-period dynamic multiplier (etc.)= effect of
change in Xt–2 on Yt, holding constant Xt, Xt–1, Xt–3,…
 Cumulative dynamic multipliers
o Ex: the 2-period cumulative dynamic multiplier
= 1 + 2 + 3 (etc.)
23-8
Exogeneity in time series regression
Exogeneity (past and present)
X is exogenous if E(ut|Xt,Xt–1,Xt–2,…) = 0.
Strict Exogeneity (past, present, and future)
X is strictly exogenous if E(ut|…,Xt+1,Xt,Xt–1, …) = 0
 Strict exogeneity implies exogeneity
 For now we suppose that X is exogenous – we’ll return
(briefly) to the case of strict exogeneity later.
 If X is exogenous then OLS estimates the dynamic
causal effect on Y of a change in X. Specifically,…
23-9
Estimation of Dynamic Causal Effects with
Exogenous Regressors
(SW Section 13.3)
Yt = 0 + 1Xt + … + r+1Xt–r + ut
The Distributed Lag Model Assumptions
1. E(ut|Xt,Xt–1,Xt–2,…) = 0 (X is exogenous)
2. (a) Y and X have stationary distributions;
(b) (Yt,Xt) and (Yt–j,Xt–j) become independent as j
gets large
3. Y and X have eight nonzero finite moments
4. There is no perfect multicollinearity.
23-10
 Assumptions 1 and 4 are familiar
 Assumption 3 is familiar, except for 8 (not four) finite
moments – this has to do with HAC estimators
 Assumption 2 is different – before it was (Xi,Yi) are
i.i.d. – this now becomes more complicated.
2. (a) Y and X have stationary distributions;
 If so, the coefficients don’t change within the
sample (internal validity);
 and the results can be extrapolated outside the
sample (external validity).
 This is the time series counterpart of the
“identically distributed” part of i.i.d.
23-11
2. (b) (Yt,Xt) and (Yt–j,Xt–j) become independent as j
gets large
 Intuitively, this says that we have separate
experiments for time periods that are widely
separated.
 In cross-sectional data, we assumed that Y and X
were i.i.d., a consequence of simple random
sampling – this led to the CLT.
 A version of the CLT holds for time series
variables that become independent as their
temporal separation increases – assumption 2(b)
is the time series counterpart of the
“independently distributed” part of i.i.d.
23-12
Under the Distributed Lag Model Assumptions:
 OLS yields consistent estimators of 1, 2,…,r (of
the dynamic multipliers)
 The sampling distribution of ˆ1 , etc., is normal
 However, the formula for the variance of this
sampling distribution is not the usual one from crosssectional (i.i.d.) data, because ut is not i.i.d. – it is
serially correlated.
 This means that the usual OLS standard errors (usual
STATA printout) are wrong
 We need to use, instead, SEs that are robust to
autocorrelation as well as to heteroskedasticity…
23-13
Heteroskedasticity and Autocorrelation-Consistent
(HAC) Standard Errors
(SW Section 13.4)
 When ut is serially correlated, the variance of the
sampling distribution of the OLS estimator is
different.
 Consequently, we need to use a different formula for
the standard errors.
 This is easy to do using STATA and most (but not all)
other statistical software.
23-14
The math…
Consider the case of no lags:
Yt = 0 + 1Xt + ut
“Recall” that the OLS estimator is:
1 T
( X t  X )(Yt  Y )

T t 1
ˆ
1 =
1 T
2
(
X

X
)

t
T t 1
so..
23-15
1 T
( X t  X )ut

T t 1
ˆ
1 – 1 =
1 T
2
(
X

X
)
 t
T t 1
(this is SW App. 4.3)
1 T
( X t  X )ut

T t 1
=
1 T
2
(
X

X
)

t
T t 1
so
ˆ1 – 1
1 T
vt

T t 1

2
X
in large samples
where vt = (Xt – X )ut (this is still SW App. 4.3)
23-16
ˆ1 – 1
1 T
vt

T t 1

2
X
in large samples
so, in large samples,
T
1
var( ˆ1 ) = var(  vt )/ ( X2 )2 (still SW App. 4.3)
T t 1
What happens with time series data? Consider T = 2:
1 T
var(  vt ) = var[½(v1+v2)]
T t 1
= ¼[var(v1) + var(v2) + 2cov(v1,v2)]
23-17
so
1 2
var(  vt ) = ¼[var(v1) + var(v2) + 2cov(v1,v2)]
2 t 1
= ½ v2 + ½1 v2
(1 = corr(v1,v2))
= ½ v2 f2, where f2 = (1+1)
 In i.i.d. (cross-section) data, 1 = 0 so f2 = 1 – which
gives the usual formula for var( ˆ1 ).
 In time series data, 1
0, so var( ˆ1 ) is not given by
the usual formula
 Conventional OLS SE’s are wrong when ut is serially
correlated (STATA printout is wrong).
Expression for var( ˆ1 ), general T
23-18
1 T
 v2
var(  v ) =
T t 1
T
fT
so
2


1

v
ˆ
var( 1 ) = 
fT
2 2
 T ( X ) 
where
T 
fT = 1  2  
T
j 1 
T 1
j
j

The OLS SEs are off by the factor fT (which can be big!)
23-19
HAC Standard Errors
 Conventional OLS SEs (heteroskedasticity-robust or
not) are wrong when there is autocorrelation
 So, we need a new formula that produces SEs that are
robust to autocorrelation as well as heteroskedasticity
We need Heteroskedasticity and AutocorrelationConsistent (HAC) standard errors
 If we knew the factor fT, we could just make the
adjustment.
 But we don’t know fT – it depends on unknown
autocorrelations.
 HAC SEs replace fT with an estimator of fT
23-20
HAC SEs, ctd.
2
T 1


1

T 
v
ˆ
var( 1 ) = 
fT , where fT = 1  2 
2 2
T
j 1 
 T ( X ) 
j
j

The most commonly used estimator of fT is:
ˆf = 1  2  m  j  

T

 j
m 
j 1 
m 1
 fˆt sometimes called “Newey-West” weights
  j is an estimator of j
 m is called the truncation parameter
 What truncation parameter to use in practice?
o Use the Goldilocks method
o Or, try m = 0.75T1/3
23-21
Example: OJ and HAC estimators in STATA
.
.
.
.
.
.
gen
gen
gen
gen
gen
gen
l1fdd
l2fdd
l3fdd
l4fdd
l5fdd
l6fdd
=
=
=
=
=
=
L1.fdd;
L2.fdd;
L3.fdd;
L4.fdd;
L5.fdd;
L6.fdd;
generate lag #1
generate lag #2
.
.
.
. reg dlpoj l1fdd if tin(1950m1,2000m12), r;
Regression with robust standard errors
NOT HAC SEs
Number of obs
F( 1,
610)
Prob > F
R-squared
Root MSE
=
=
=
=
=
612
3.97
0.0467
0.0101
5.0438
-----------------------------------------------------------------------------|
Robust
dlpoj |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------l1fdd |
.1529217
.0767206
1.99
0.047
.0022532
.3035903
_cons | -.2097734
.2071122
-1.01
0.312
-.6165128
.196966
------------------------------------------------------------------------------
Example: OJ and HAC estimators in STATA, ctd.
23-22
Example: OJ and HAC estimators in STATA, ctd
Now compute Newey-West SEs:
. newey dlpoj l1fdd if tin(1950m1,2000m12), lag(8);
Regression with Newey-West standard errors
maximum lag : 8
Number of obs
F( 1,
610)
Prob > F
=
=
=
612
3.83
0.0507
-----------------------------------------------------------------------------|
Newey-West
dlpoj |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------l1fdd |
.1529217
.0781195
1.96
0.051
-.000494
.3063375
_cons | -.2097734
.2402217
-0.87
0.383
-.6815353
.2619885
-----------------------------------------------------------------------------Uses autocorrelations up to m=8 to compute the SEs
rule-of-thumb: 0.75*(6121/3) = 6.4
8, rounded up a little.
OK, in this case the difference is small, but not always so!
23-23
Example: OJ and HAC estimators in STATA, ctd.
. global lfdd6
.
"fdd l1fdd l2fdd l3fdd l4fdd l5fdd l6fdd";
newey dlpoj $lfdd6 if tin(1950m1,2000m12), lag(7);
Regression with Newey-West standard errors
maximum lag : 7
Number of obs
F( 7,
604)
Prob > F
=
=
=
612
3.56
0.0009
-----------------------------------------------------------------------------|
Newey-West
dlpoj |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------fdd |
.4693121
.1359686
3.45
0.001
.2022834
.7363407
l1fdd |
.1430512
.0837047
1.71
0.088
-.0213364
.3074388
l2fdd |
.0564234
.0561724
1.00
0.316
-.0538936
.1667404
l3fdd |
.0722595
.0468776
1.54
0.124
-.0198033
.1643223
l4fdd |
.0343244
.0295141
1.16
0.245
-.0236383
.0922871
l5fdd |
.0468222
.0308791
1.52
0.130
-.0138212
.1074657
l6fdd |
.0481115
.0446404
1.08
0.282
-.0395577
.1357807
_cons | -.6505183
.2336986
-2.78
0.006
-1.109479
-.1915578
----------------------------------------------------------------------------- global lfdd6 defines a string which is all the additional lags
 What are the estimated dynamic multipliers (dynamic effects)?
23-24
Do I need to use HAC SEs when I estimate an AR or
an ADL model?
NO.
 The problem to which HAC SEs are the solution arises
when ut is serially correlated
 If ut is serially uncorrelated, then OLS SE’s are fine
 In AR and ADL models, the errors are serially
uncorrelated if you have included enough lags of Y
o If you include enough lags of Y, then the error term
can’t be predicted using past Y, or equivalently by
past u – so u is serially uncorrelated
23-25
Estimation of Dynamic Causal Effects with Strictly
Exogenous Regressors
(SW Section 13.5)
 X is strictly exogenous if E(ut|…,Xt+1,Xt,Xt–1, …) = 0
 If X is strictly exogenous, there are more efficient
ways to estimate dynamic causal effects than by a
distributed lag regression.
o Generalized Least Squares (GLS)
o Autoregressive Distributed Lag (ADL)
 But the condition of strict exogeneity is very strong,
so this condition is rarely plausible in practice.
 So we won’t cover GLS or ADL estimation of
dynamic causal effects (Section 13.5 is optional)
23-26
Analysis of the OJ Price Data
(SW Section 13.6)
What is the dynamic causal effect (what are the dynamic
multipliers) of a unit increase in FDD on OJ prices?
%ChgPt = 0 + 1FDDt + … + r+1FDDt–r + ut
 What r to use?
How about 18? (Goldilocks method)
 What m (Newey-West truncation parameter) to use?
m = .75 6121/3 = 6.4 7
23-27
23-28
23-29
23-30
23-31
These dynamic multipliers were estimated using a
distributed lag model. Should we attempt to obtain more
efficient estimates using GLS or an ADL model?
 Is FDD strictly exogenous in the distributed lag
regression?
%ChgPt = 0 + 1FDDt + … + r+1FDDt–r + ut
 OJ commodity traders can’t change the weather.
 So this implies that corr(ut,FDDt+1) = 0, right?
23-32
When Can You Estimate Dynamic Causal Effects?
That is, When is Exogeneity Plausible?
(SW Section 13.7)
In the following examples,
 is X exogenous?
 is X strictly exogenous?
Examples:
1. Y = OJ prices, X = FDD in Orlando
2. Y = Australian exports, X = US GDP (effect of US
income on demand for Australian exports)
23-33
Examples, ctd.
3. Y = EU exports, X = US GDP (effect of US income on
demand for EU exports)
4. Y = US rate of inflation, X = percentage change in
world oil prices (as set by OPEC) (effect of OPEC oil
price increase on inflation)
5. Y = GDP growth, X =Federal Funds rate (the effect of
monetary policy on output growth)
6. Y = change in the rate of inflation, X = unemployment
rate on inflation (the Phillips curve)
23-34
Exogeneity, ctd.
 You must evaluate exogeneity and strict exogeneity
on a case by case basis
 Exogeneity is often not plausible in time series data
because of simultaneous causality
 Strict exogeneity is rarely plausible in time series data
because of feedback.
23-35
Estimation of Dynamic Causal Effects: Summary
(SW Section 13.8)
 Dynamic causal effects are measurable in theory using
a randomized controlled experiment with repeated
measurements over time.
 When X is exogenous, you can estimate dynamic causal
effects using a distributed lag regression
 If u is serially correlated, conventional OLS SEs are
incorrect; you must use HAC SEs
 To decide whether X is exogenous, think hard!
23-36